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The pre-Bötzinger complex is a neuronal network with excitatory coupling, which participates in modulation of respiratory rhythms via the generation of complex firing rhythm patterns and synchronization transitions of rhythm patterns. In the present paper, a mathematical model of single neuron that exhibits complex transition processes from bursting to spiking is selected as a unit, the network model of the pre-Bötzinger complex composed of two neurons with excitatory coupling is constructed, multiple synchronous rhythm patterns and complex transition processes of the synchronous rhythm patterns related to the biological experimental observations are simulated, and the corresponding bifurcation mechanism is acquired with the fast-slow variable dissection method. When the initial values of two neurons of the pre-Bötzinger complex are the same, with increasing the excitatory coupling strength, the theoretical model of the pre-Bötzinger complex shows complete synchronization transition processes from "fold/homoclinic" bursting, to "subHopf/subHopf" bursting, and at last to period-1 spiking. When the initial values are different, with the increases of the excitatory coupling intensity, the rhythm transition processes begin from phase synchronization behaviors including "fold/homoclinic" bursting, "fold/fold limit cycle" bursting, mixed bursting composed of "subHopf/subHopf" bursting and "fold/fold limit cycle" bursting, and "subHopf/ subHopf" bursting in sequence, and to anti-phase synchronous behavior of the period-1 spiking. The complete (in-phase) synchronous period-1 spiking for the same initial values exhibits bifurcation mechanism different from the anti-phase synchronous period-1 spiking for different initial values. The anti-phase synchronous period-1 spiking presents a novel and abnormal example of the synchronization at large excitatory coupling strength, which is different from the traditional viewpoint that large excitatory coupling often induces in-phase synchronous behavior. The results present the synchronization transition process and complex bifurcation mechanism from bursting to period-1 spiking of the pre-Bötzinger complex, and the abnormal synchronization example enriches the contents of nonlinear dynamics.
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参数 参数值 参数 参数值 参数 参数值 参数 参数值 C 21 pF $ {\sigma _{ {\rm{m_p} }} } $ –6 mV $ {g_{ {\rm{Nap} }} } $ 2.8 nS ${E_{{\rm{Na}}}}$ 50 mV $ {\theta _{ {\rm{m_p} }} } $ –40 mV ${\sigma _{\rm{m}}}$ –5 mV ${g_{{\rm{Na}}}}$ 28 nS ${E_{\rm{K}}}$ –85 mV ${\theta _{\rm{m}}}$ –34 mV $\sigma {}_{\rm{h}}$ 6 mV ${g_{\rm{L}}}$ 2.8 nS ${E_{\rm{L}}}$ –65 mV ${\theta _{\rm{h}}}$ –48 mV ${\sigma _{\rm{n}}}$ –4 mV ${g_{ {\text{tonic-e} } } }$ 0.4 nS ${\bar \tau _{\rm{h}}}$ 10000 ms ${\theta _{\rm{n}}}$ –29 mV ${\sigma _{\rm{s}}}$ –5 mV ${\varepsilon _{}}$ 6 ${\bar \tau _{\rm{n}}}$ 5 ms $\theta {}_{\rm{s}}$ –10 mV ${\alpha _{\rm{s}}}$ –5 mV 
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关键点 h的值 F1 F2 subh HC LPC 共存区域 $ {g_{\rm{K} }} = 7.1\;{\rm{nS}} $ 0.4928 –1.6780 0.2128 0.3265 0.4308 [0.3265, 0.4308] $ {g_{\rm{K} }} = 7.8\;{\rm{nS}} $ 0.4928 –1.6680 0.2858 0.3476 0.4973 [0.3476, 0.4928] $ {g_{\rm{K} }} = 10.0 \;{\rm{nS}} $ 0.4928 –1.6390 0.5072 0.3941 0.7025 [0.3941, 0.4928] $ {g_{\rm{K} }} = 25.0 \;{\rm{nS}} $ 0.4928 –1.4800 1.7880 0.4849 1.9240 [0.4849, 0.4928] DownLoad:
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关键点 h的值 $g_\text{syn-e}$ = 0.35 nS $g_\text{syn-e}$ = 2.5 nS $g_\text{syn-e}$ = 5.0 nS $g_\text{syn-e}$ = 18.0 nS F1 0.4874 0.4918 0.4908 0.4856 F2 –1.6695 –1.6759 –1.6685 –1.7212 subh1 0.2817 0.2565 0.2259 0.0746 subh2 0.2858 0.2852 0.2274 0.0794 LPC1 0.4927 0.4273 0.3598 0.0960 LPC2 \ 0.3103 0.2406 –0.2504 LPC3 \ \ \ 0.0890 LPC4 \ \ \ –0.099 HC 0.3398 \ \ \ 共存区域 [0.3398, 0.4927] [0.3103, 0.4273] [0.2406, 0.3598] [0.0960, 0.250]和[0.0890, 0.099] DownLoad:
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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56]
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